It is known that the class of all graphs not containing a graph H as an induced subgraph is cop-bounded if and only if H is a forest whose every component is a path [4]. In this paper, we characterize all sets $\mathcal{H}$ of graphs with bounded diameter, such that $\mathcal{H}$-free graphs are cop-bounded. This, in particular, gives a characterization of cop-bounded classes of graphs defined by a finite set of connected graphs as forbidden induced subgraphs. Furthermore, we extend our characterization to the case of cop-bounded classes of graphs defined by a set $\mathcal{H}$ of forbidden graphs such that the components of members of $\mathcal{H}$ have bounded diameter.

中文翻译：

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具有一组禁止的诱导子图的图上的警察和强盗

众所周知，当且仅当H是其每个组成部分都是路径的森林时，才将不包含图H作为诱导子图的所有图的类称为cop-bound [4]。在本文中，我们表征了所有集合$\mathcal{H}$ 有界直径的图 $\mathcal{H}$-free图受警察限制。尤其是，这给出了由有限的一组连通图定义的警察约束图类别的表征，作为禁止的诱导子图。此外，我们将特征描述扩展到由集合定义的图约束的图类的情况$\mathcal{H}$ 禁止图的组成 $\mathcal{H}$ 直径有界。